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On the Geographic Incentives of Multiple Concurrent Proposers
Kushal Babel, Jason Milionis, Eric Xue
TL;DR
The paper asks whether multiple concurrent proposers incentivize geographic decentralization, given that users consider both proposer proximity and quorum-propagation latency. It models proposer location choices around total time to inclusion and finds that quorum-radius incentives produce co-location when proposers also attest, but can promote diversity when roles are separate or quorum-radius influence is weakened.
Problem
It is unclear whether multiple concurrent proposers make users choose the closest proposer and thereby incentivize geographic dispersion.
Source: §1 · p. 2Method
The paper models proposers maximizing market share when users compare user-to-proposer distance plus the proposer’s quorum radius.
Source: §2 · p. 7, §3 · p. 10Results
Quorum-radius incentives yield co-location when validators also attest, while role separation, preconfirmations, and faster intervalidator connections can promote geographic diversity.
Source: §1 · p. 2, §1.4 · pp. 4–5Takeaways & Limitations
MCP alone may not change co-location incentives; geographic spread can depend on role separation or weakening quorum-radius influence.
Source: §1 · p. 2, §1.5 · p. 5The paper leaves unresolved whether Sybil participation in both roles gives a meaningful proposer advantage, especially when the attester set is large.
Source: §1.4 · p. 4
Abstract
from arXiv · showhide
We study whether systems with multiple concurrent proposers (MCP) incentivize the geographic decentralization of proposers. In our model, concurrent proposers strategically choose their locations to maximize their captured market share of users, who decide where to send transactions by comparing each proposer's time to inclusion. The time to inclusion between a user and a proposer consists of two terms: the distance between them, and the distance between the proposer and her nearest quorum of attesters, the latter of which we term the quorum radius. We formally prove that proposers' quorum radii are the determinant of their incentives: proposers optimally choose where to locate in order to minimize their quorum radii. We obtain as a result co-location when proposers also serve as attesters, but geographic diversity when these roles are separate. We also examine pre-confirmations and faster intervalidator connections, showing that they attenuate the influence of the quorum radius and further destabilize co-location incentives.
1 Introduction
The paper asks whether multiple concurrent proposers promote geographic decentralization, since users’ choices depend on both proposer proximity and quorum-propagation latency. It finds that quorum-radius incentives can drive co-location when proposers also attest, while role separation and weaker quorum-radius influence can promote geographic diversity.
1 Introduction
- Geographic decentralization is studied for resilience to regional failures, proximity to users, and independence from local decisions, despite validators’ concentration in the U.S. and Europe.The paper situates its question amid recent work on blockchain geographic topology [MGZJ26, MJ23, RMBK26, YÖWZ26, ÃWC+26].Source: §1 · p. 1
- The paper asks whether MCP necessarily makes users choose the closest proposer and the closest proposer capture each opportunity; it answers both questions no.The conventional argument assumes users prefer proximity and location-game incentives then drive validators to spread out.Source: §1 · p. 2
1.1 Model
- The model has users and validators on a line, with users choosing the validator minimizing user distance plus the validator’s distance to a quorum.The sum represents the time from transaction transmission until a quorum of validators receives it; validators compete to maximize market share.Source: §1.1 · pp. 2–3
1.2 Extensions
- Preconfirmation users choose the closest validator, while faster intervalidator links reduce the quorum-latency component of users’ inclusion-time comparisons.The model also considers retail users who broadcast transactions to validators competing to include them first.Source: §1.2 · pp. 3–4, §1.1 · p. 3
1.3 Attester-Proposer Separation
- With separate proposer and attester sets, users compare their distance to a proposer with that proposer’s distance to its nearest attester quorum.The attester distribution is exogenous in this model.Source: §1.3 · p. 4
1.4 Results
- Validators co-locate in the basic model, while preconfirmations and faster connections weaken co-location and separated attesters can promote proposer dispersion.With uniformly distributed attesters, separated proposers locate at attester-quorum midpoints; the covering radius decreases as preconfirmation use, connection speed, or proposer count increases.Source: §1.4 · pp. 4–5
1.5 Practical Takeaways
- The practical conclusions are that MCP alone may not change co-location, while location incentives or sufficiently strong preconfirmations and faster links can promote dispersion.When proposers and attesters are separate, proposers tend to mirror attesters’ geographic distribution; spreading attesters can therefore encourage proposer spread.Source: §1.5 · p. 5
1.6 Related Work
- Unlike classic Hotelling competition, intervalidator distance also enters user preferences; the paper’s separation model endogenously restricts proposer locations before Hotelling dynamics emerge.The authors distinguish their model from prior MCP work by including quorum propagation and allowing late transactions to enter a later block [ÃWC+26].Source: §1.6.1 · p. 6, §1.6.2 · p. 6
2 Model
The model treats proposers as choosing locations to maximize market share, with user choice based on total inclusion latency: distance to the proposer plus quorum radius. It specifies distinct user types and network extensions, then defines equilibrium in pure and mixed strategies.
2 Model
- Users are distributed along the unit interval, and proposers choose locations to maximize market share based on total latency, the sum of user-to-proposer distance and quorum radius.The quorum radius is the time to reach a quorum of attesters, which may be the other proposers or a separate node set.Source: §2 · pp. 6–7
2.1 Sophisticated Users
- Sophisticated users send as late as possible by choosing a proposer that minimizes total latency to meet their target block deadline.Their market shares aggregate the users assigned to each latency-minimizing proposer under the user distribution.Source: §2.1 · pp. 7–8
2.2 Retail Users
- Retail users send when transactions are generated, and proposers capture transactions they can receive and propagate before a block deadline.As block time approaches zero, the retail model’s market shares converge to those of the sophisticated-user model.Source: §2.2 · p. 8
2.3 Preconfirmations
- Preconfirmation users choose based on latency to a promise of future inclusion, while the remaining α fraction choose based on time to inclusion.When α = 0, the model recovers the classic Hotelling model.Source: §2.3 · p. 9
2.4 Faster Intervalidator Connections
- Dedicated infrastructure scales quorum-propagation latency by µ, changing total latency to user distance plus µ times the quorum radius.The model assumes all proposers use the dedicated network and leaves the rest of the model unchanged.Source: §2.4 · p. 9
2.5 Equilibrium
- A pure equilibrium has no proposer able to increase market share by changing location, while a mixed equilibrium allows independent randomized locations.For continuous attester distributions, the quorum-radius definition uses an infimum because a nearest quorum may not exist.Source: §2 · p. 7, §2.5 · p. 9
3 Results
Proposers’ incentives turn on minimizing quorum radius, but the equilibrium locations depend on whether proposers also attest and on how strongly quorum radius shapes user choices. Proposers co-locate when they also attest, whereas separate attesters can induce midpoint locations and weaker quorum-radius influence can destabilize co-location.
3.1 Minimizing Quorum Radius Dictates Proposer Locations
- Reducing a proposer’s quorum radius weakly reduces her total latency for all users and strictly reduces it for those she moves closer to, increasing market share if peers’ radii do not fall.Section 3.1 considers no preconfirmation users and intervalidator connections no faster than the public internet.Source: §3.1 · p. 10
3.2 Co-location
- With n ≥ 4 and full-support users, co-location is the unique pure equilibrium when proposers also serve as attesters.A quorum-radius-zero cluster weakly dominates a lone deviator; the remaining split configurations also admit a profitable deviation.Source: §3.2 · pp. 10–11
3.3 Attester-Proposer Separation
- With uniform attesters and sufficient user-tail mass, proposers locate only at attester-quorum midpoints, where equal quorum radii make routing depend only on proposer latency.The resulting pure equilibria coincide with those of a Hotelling game restricted to those midpoints; the result extends to attester quorums with equal diameters.Source: §3.3 · p. 12
3.4 Weakening Influence of Quorum Radius Incentives Dispersion: Faster Intervalidator Connections and Preconfirmations
- Faster intervalidator connections and preconfirmations weaken quorum-radius influence, favoring closer proposers and destabilizing co-location.With faster connections, a user between a lone proposer and a cluster prefers the lone proposer because the final leg uses the faster link.Source: §3.4 · pp. 12–13
- For sophisticated users, co-location fails with faster intervalidator connections when µ < 1−2/n, or with preconfirmation users when α < 1−1/(n−1).These thresholds assume n ≥ 4, full-support user distributions without point masses, and the respective connection or preconfirmation settings.Source: §3.4 · p. 13
AI Usage
The paper frames concurrent-proposer location as a spatial competition problem in which inclusion latency combines user-to-proposer distance with quorum radius. It relates this model to prior spatial-competition work while distinguishing its results from those prior models.
AI Usage
- The authors used Claude and ChatGPT for examples, mathematical checks, literature review, and figures; they attribute the exposition, model, results, and proofs to themselves.They state that they take full responsibility for the paper’s content.Source: AI Usage · p. 13
A.1 Hotelling Spatial Competition
- The model’s key insight is that proposers’ restrictions to particular locations arise endogenously and induce Hotelling dynamics, so existing Hotelling results do not imply its equilibrium characterizations.The paper notes that prior work focuses on large markets, duopolies, or price of anarchy.Source: §A.1 · p. 19
A.2 Attractiveness and Other Firms
- In comparison-shopping models, consumers may favor a group of nearby firms, but those models’ trip-cost structure differs from validators’ min-max propagation latency.The paper says the degree to which those models’ insights apply to its problem is unclear.Source: §A.2 · p. 19
A.3 Attractiveness and Fixed Spatial Features
- Even with uncountably many pseudo-targets, proposers restrict themselves to attester-quorum midpoints when all quorums have the same diameter.The cited position-optimization result does not imply this finding because its required conditions fail in this setting.Source: §A.3 · p. 21
A.4 Geographic Decentralization
- Prior work proposes geographic-diversity incentives and penalty exemptions for isolated validators, while another study finds co-location incentives under Ethereum’s single-proposer design.That study recommends reducing monopoly power and reward sensitivity to timing advantages, and diversifying suppliers, relays, builders, and information sources.Source: §A.4 · p. 21
B Quorum Radius
- A user’s inclusion latency combines distance to the proposer with μ times the proposer’s distance to its m-th nearest peer, the latter capturing propagation latency.The paper calls the proposer-to-peer distance component the quorum radius; μ represents faster intervalidator connections.Source: B Quorum Radius · p. 21
B.1 Some Structural Lemmas
The structural lemmas characterize users’ preferred-proposer sets and how distinct proposers’ market shares relate under specified conditions.
B.1 Some Structural Lemmas
- Each proposer’s set of latency-minimizing users is a closed interval containing that proposer’s location.If no validators lie between two proposers, their user sets intersect.Source: §B.1 · p. 22
- When μ < 1, a user located at proposer i’s position prefers only proposers co-located with i.This is the stated characterization of S(vi, v).Source: §B.1 · p. 22
- When μ < 1, distinct proposer locations have disjoint interiors of latency-minimizing user sets, so their market shares essentially partition the unit interval.This partition claim concerns interiors; it does not exclude boundary ties.Source: §B.1 · p. 23
B.2 Minimizing Quorum Radius Is Profitable
The paper establishes that reducing one’s quorum radius without increasing opponents’ radii is weakly profitable. In the specified dual-role setting, pure equilibrium requires co-location.
B.2 Minimizing Quorum Radius Is Profitable
- A proposer who can reduce her quorum radius without reducing opponents’ radii weakly increases her market share.The paper’s lemma also specifies conditions under which the gain is strict.Source: §B.2 · p. 24
C Co-location
- With n ≥ 4 and a full-support user distribution, a pure equilibrium exists if and only if all validators are co-located.The result is stated for validators that both attest and propose, with α = μ = 1.Source: C Co-location · p. 27
C.1 Proof Overview
The proof establishes co-location as an equilibrium everywhere and then rules out other pure equilibria through profitable deviations.
C.1 Proof Overview
- To show all pure equilibria are co-located, the proof rules out most configurations through profitable local deviations, then eliminates the remaining clustered configurations by a market-share computation.The remaining configurations have n − m + 1 validators at each of two points and the others at the midpoint.Source: §C.1 · p. 27
C.2.1 Sufficiency Of Co-location
- Co-location at any location is an equilibrium in every metric space.The proof follows from the triangle inequality.Source: §C.1 · p. 27, §C.2.1 · p. 28
C.2.2 Necessity Of Co-location
This section examines the necessity of co-location in pure equilibrium, establishing that with at least four validators and a full-support user distribution, all validators must co-locate because a validator can profitably reduce its quorum radius without worsening opponents’ radii or passing them. It rules out a candidate equilibrium via a profitable deviation and considers how faster intervalidator connections weaken co-location and improve validator coverage under the stated conditions.
C.2.2 Necessity Of Co-location
- Each validator’s user interval is closed and contains her location; if no validators lie between two validators, their user intervals are disjoint.Source: §C.2.2 · p. 28
- Under full support, if a validator can reduce her quorum radius without reducing opponents’ radii or passing them, the deviation strictly increases her market share.This is the profitable-deviation condition used to rule out separated equilibrium configurations.Source: §C.2.2 · p. 28
- At a pure equilibrium with full-support users, an m-sized set of co-located validators implies that all validators are co-located.Any validator outside that group would have a profitable deviation under the preceding condition.Source: §C.2.2 · p. 30
- For n = 4, two validators at each of two distinct locations cannot form an equilibrium, even without full support of the user distribution.The proof shows that at least one validator can profit by moving to the other location.Source: §C.2.2 · p. 35
- With n ≥ 4 and full-support users, every pure-equilibrium configuration has all validators co-located.The theorem states a necessary condition for pure equilibrium.Source: §C.2.2 · p. 37
- When v_{n−m+1}=v_m, validator m can strictly profit by moving from v_m to v_m+ε, contradicting equilibrium.The deviation reduces validator m’s quorum radius while not increasing the radii of validators strictly to the right of v_m.Source: §C.2.2 · p. 39
D Faster Intervalidator Connections
- Validator coverage improves as connections speed up and validator count rises, and as block time falls, but only to a certain extent.These trends are stated for validator location decisions when validators simultaneously serve as attesters and proposers.Source: D Faster Intervalidator Connections · p. 40
- For n ≥ 4, co-location is not an equilibrium when µ < 1 − 2/n for any ∆ > 0, or when sophisticated users face µ < 1.For uniformly distributed users, the theorem also bounds expected distance to the nearest validator at any mixed equilibrium, but the supplied passage omits the bound.Source: D Faster Intervalidator Connections · p. 40
D.1 Proof Overviews
Faster intervalidator connections can make co-location unattractive by shifting users toward closer validators. At equilibrium, this shift also improves market coverage by limiting how far users are from their closest validators.
D.1 Proof Overviews
- When µ is sufficiently small, co-location is no longer an equilibrium because a validator can become the fastest gateway for an entire side of the market by moving away from peers.Users then prefer closer validators.Source: §D.1 · p. 40
- Faster connections encourage validators to cover the market when users are uniformly distributed, since a validator may otherwise gain share by moving closer to underserved users.At equilibrium, even the validator with the smallest market share does not want to make that move, which bounds the distance to users’ closest validators.Source: §D.1 · p. 40
- The coverage argument uses the fact that a validator can attract a user and surrounding market when that user’s closest validators are far away.The no-deviation condition at equilibrium is what yields the distance bound.Source: §D.1 · p. 40
D.2 Faster Intervalidator Connections: Instability of Co-location
With sufficiently fast intervalidator connections, co-location is not an equilibrium under the stated user-distribution conditions. The threshold depends on whether users compare inclusion time continuously or in blocks.
D.2 Faster Intervalidator Connections: Instability of Co-location
- For n ≥ 4, full-support user distributions without point masses rule out co-location when µ < 1 in the sophisticated-user model or µ < 1 − 2/n in the block-time model.The block-time threshold applies for any block time ∆ > 0.Source: §D.2 · p. 41
- A validator can profitably deviate from co-location by moving a small distance toward one side of the market, where her market share approaches at least half.For the other side of the market, the argument considers a deviation in the opposite direction.Source: §D.2 · p. 41
- In the block-time model, a small relocation changes the relative inclusion latencies across user intervals, yielding a market-share gain under the stated threshold.The proof analyzes a deviation of ε ≪ ∆ and establishes the result for either side of the co-location point.Source: §D.2 · pp. 41–42
D.3 Faster Intervalidator Connections: Coverage
Faster connections bound the distance between users and nearby validators at mixed equilibrium. Preconfirmations have a similar effect on validator locations, with a distinct bound and no dependence on block time.
D.3 Faster Intervalidator Connections: Coverage
- At mixed equilibrium, faster connections bound the distance from each location to its closest validators; for uniformly distributed users, the paper derives an explicit bound.The argument uses guaranteed market share on either side of a location when nearby peers are more than ε away, then bounds the probability of such gaps.Source: §D.3 · pp. 42, 44
- If a location is more than ε from every other validator, a validator relocating there can secure market share from users on both sides.This guarantee underlies the equilibrium bound on how often users’ closest validators can be far away.Source: §D.3 · p. 42
E Preconfirmations
- Preconfirmations have a location effect similar to faster connections, but the resulting bound on user-to-validator distance differs and does not depend on block time.The theorem rules out co-location for α < 1 when the user distribution has full support and no point masses, for all block times; for sophisticated users, it requires α < 1 − 1/(n−1).Source: E Preconfirmations · p. 45
E.1 Proof Overviews
Preconfirmations make co-location unstable because a proposer can move slightly and capture a constant fraction of nearby preconfirmation users while retaining some time-to-inclusion users. In the block-time model, any positive fraction of preconfirmation users suffices for this instability.
E.1 Proof Overviews
- A proposer can move just beside co-located peers, retain Ω(1/n) time-to-inclusion users, and capture all preconfirmation users on one side.That side's preconfirmation users form a constant fraction of the market.Source: §E.1 · p. 46
- The coverage proof uses the fact that a validator can capture nearby preconfirmation users by relocating, then bounds how far users may be from their nearest validator.The argument proceeds from equilibrium incentives to tail bounds and then to expected distance.Source: §E.1 · p. 46
E.2 Preconfirmations: Instability of Co-location
- With full-support, atomless user distributions and n ≥ 4, co-location is not an equilibrium under the theorem's stated condition.The supplied theorem statement truncates before displaying that condition.Source: §E.2 · p. 46
- In the block-time model, α < 1 for any Δ > 0, so a positive fraction of users are preconfirmation users.A sufficiently small relocation raises a proposer's market share above 1/n; the argument applies on either side of the co-located position.Source: §E.2 · p. 46
E.3 Preconfirmations: Coverage
When α < 1, equilibrium incentives constrain gaps in validator coverage: a validator can profit by relocating to an uncovered location and capturing nearby preconfirmation users.
E.3 Preconfirmations: Coverage
- If α < 1 and a location u is more than ε from every other validator, validator i becomes uniquely closest to users in [u ± ε/2] ∩ [0,1].Source: §E.3 · pp. 47–48
- For mixed equilibria with α < 1, the probability that no proposer lies within ε of any location is bounded in terms of nearby user mass.For uniform users, the result has a more explicit bound; the supplied passages do not provide its complete expression.Source: §E.3 · p. 48
- The coverage argument begins by comparing a proposer’s expected equilibrium market share with the share she can guarantee by deviating to a location with no nearby competitors.The weakest-expected-share proposer has expected share at most 1/n.Source: §E.3 · p. 48
F Attester-Proposer Separation
Separating attesters from proposers changes the location incentives: under a regularity condition, nearly all proposers locate at attester-quorum midpoints, with possible exceptions at the boundaries. Equal quorum radii at those midpoints reduce user choice to a Hotelling game.
F Attester-Proposer Separation
- With disjoint attesters and proposers, all but possibly the leftmost and rightmost proposers locate at attester-quorum midpoints in pure equilibrium.The theorem assumes full support for users and equal diameters of attester quorums; any departures are confined to the boundaries.Source: F Attester-Proposer Separation · p. 49
- When quorum diameters are equal, midpoint quorum radii coincide, so users choose among proposers based only on latency to proposers.The resulting location game is a pure Hotelling game over quorum midpoints, which has spread-out equilibria.Source: F Attester-Proposer Separation · p. 49
- For uniform attesters and uniformly distributed continuous users, proposers must locate at quorum midpoints at equilibrium, and equilibria are Hotelling equilibria over those midpoints.The section considers both discrete and continuous uniform attester distributions.Source: F Attester-Proposer Separation · p. 49
- The model has proposers send transactions to a quorum of attesters distributed on the unit interval, with quorum radius determined by attester locations.The section's quorum uses more than two-thirds of the attester measure.Source: F Attester-Proposer Separation · p. 49
F.1 A Regularity Condition
This section characterizes quorum-radius minimizers under a regularity condition and uses them to reduce proposer location choices to a Hotelling game over those minimizers.
F.1 A Regularity Condition
- The quorum radius D is 1-Lipschitz, and each proposer’s latency-minimizing user set is a closed interval containing her location.If no validators lie between two proposers, their latency-minimizing user sets overlap.Source: §F.1 · p. 50
- The quorum radius at x equals the smallest maximum distance from x to the endpoints of a minimal interval containing more than two-thirds of attesters.This expresses D(x) through the minimal intervals in I∗.Source: §F.1 · p. 51
- Locations minimizing quorum radius are midpoints of minimal attester intervals; they are exactly those midpoints when all such intervals have equal diameter.The minimum set X is nonempty because D is continuous.Source: §F.1 · p. 51
- With equal diameters of minimal attester intervals, D(x) equals the minimum quorum radius plus the distance from x to its nearest minimizer.Thus, locations outside X are weakly dominated under the same condition.Source: §F.1 · p. 52
F.2 Reduction to Pure Hotelling
- Under equal interval diameters and full-support users, a proposer outside X is not best-responding when another proposer lies weakly on the same side and contains a closest point in X.This is the condition used to reduce the equilibrium analysis to a Hotelling game over X.Source: §F.2 · p. 53
- Relocating a proposer does not change other proposers’ quorum radii when attesters and proposers are separate.The proof contrasts this with the case where proposers also serve as attesters.Source: §F.2 · p. 53
F.3 Applications of Theorem F.8
The applications characterize proposer equilibria by reducing placement to Hotelling competition on locations with equal quorum radii. With uniform attesters, the available locations and equilibrium proposer counts depend on whether attesters are continuous or discrete.
Theorem F.9
- Under Theorem F.9’s assumptions, a proposer configuration is an equilibrium exactly when it is a Hotelling equilibrium restricted to X.The result assumes Δ = 0, at least four proposers, full-support user distribution, equal diameters for the relevant intervals, and user mass of at least 2/n in each tail.Source: §F.3 · p. 55
Uniform attester distributions
- With uniform attesters, proposer locations spread across the market because the quorum midpoints are spread across it; proposer counts at each midpoint track the fraction of users sending there.When the number of proposers is below the number of attesters, proposers still locate only at midpoints, but interior locations can vary.Source: §F.3 · p. 55
Theorem F.10
- With uniform users and attesters, equilibrium proposers stay within [1/3, 2/3], with endpoint occupancy and market-share conditions ruling out profitable relocations.No proposer’s market share can be smaller than another proposer’s half-market in that interval.Source: §F.3 · p. 56
Theorem F.11
- For discrete uniform attesters, equilibrium proposers occupy the quorum-midpoint set X, with endpoint and interior proposer counts constrained by the theorem’s balance conditions.When n ≥ k, a proposer can occupy each midpoint; the equilibrium conditions constrain counts at those locations.Source: §F.3 · pp. 55, 57